This paper analyses a model for combustion of a self-heating chemical (such as pool chlorine), stored in drums within a shipping container. The system is described by three coupled nonlinear differential equations for the concentration of the chemical, its temperature and the temperature within the shipping container. Self-sustained oscillations are found to occur, as a result of Hopf bifurcation. Temperature and concentration profiles are presented and compared with the predictions of a simpler two-variable approximation for the system. We study the period of oscillation and its variation with respect to the ambient temperature and the reaction parameter. Nonlinear resonances are found to exist, as the solution jumps between branches having different periods.

We show the strongly stable convergence of some non-collectively-compact approximations of compact operators. Special attention is devoted to Anselone's singularity subtraction discretization of certain singular integral operators. Numerical experiments are provided.

We consider the effect the competing mechanisms of buoyancy-driven acceleration (arising from heating a surface) and streamline curvature (due to curvature of a surface) have on the stability of boundary-layer flows. We confine our attention to vortex type instabilities (commonly referred to as Görtler vortices) which have been identified as one of the dominant mechanisms of instability in both centrifugally and buoyancy driven boundary layers. The particular model we consider consists of the boundary-layer flow over a heated (or cooled) curved rigid body. In the absence of buoyancy forcing the flow is centrifugally unstable to counter-rotating vortices aligned with the direction of the flow when the curvature is concave (in the fluid domain) and stable otherwise. Heating the rigid plate to a level sufficiently above the fluid's ambient (free-stream) temperature can also serve to render the flow unstable. We determine the level of heating required to render an otherwise centrifugally stable flow unstable and likewise, the level of body cooling that is required to render a centrifugally unstableflow stable.

A class of non-standard optimal control problems is considered. The non-standard feature of these optimal control problems is that they are of neither fixed final time nor of fixed final state. A method of solution is devised which employs a computational algorithm based on control parametrization techniques. The method is applied to the problem of maximizing the range of an aircraft-like gliding projectile with angle of attack control.

Sufficient conditions are obtained for the existence of a unique asymptotically stable periodic solution for the Lotka-Volterra two species competition system of equations when the intrinsic growth rates are periodic functions of time.

In this paper we review a simple class of fixed point models for loss networks. We illustrate how these models can readily deal with heterogeneous call types and with simple dynamic routing strategies, and we outline some of the recent mathematical advances in the study of such models. We describe how fixed point models lead to a natural and tractable definition of the implied cost of carrying a call, and how this concept is related to issues of routing and capacity expansion in loss networks.

Solutions of a homogeneous (r + 1)-term linear difference equation are given in two different forms. One involves the elements of a certain matrix, while the other is in terms of certain lower Hessenberg determinants. The results generalize some earlier results of Brown [1] for the solution of a 3-term linear difference equation.

Three stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and anindependent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.

We consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrödinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential.

In this paper we shall describe a numerical method for the solution of curve flow problems in which the normal velocity of the curve depends locally on the position, normal and curvature of the curve. The method involves approximating the curve by a number of line elements (segments) which are only allowed to move in a direction normal to the element. Hence the normal of each line element remains constant throughout the evolution. In regions of high curvature elements naturally tend to accumulate. The method easily deals with the formation of cusps as found in flame propagation problems and is computationally comparable to a naive marker particle method. As a test of the method we present a number of numerical experiments related to mean curvature flow and flows associated with flame propagation and bushfires.

A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

We consider the epidemic model with subpopulations introduced in Hethcote [5]. It is shown that if the endemic equilibrium exists, then the system is uniformly persistent. Moreover, the endemic equilibrium is globally asymptotically stable under the assumption of small effective contact rates between different subpopulations.

The present work deals with the problem of recovering a local image from localised projections using the concept of approximation identity. It is based on the observation that the Hilbert transform of an approximation identity taken from a certain class of compactly supported functions with sufficiently many zero moments has no significant spread of support. The associated algorithm uses data pertaining to the local region along with a small amount of data from its vicinity. The main features of the algorithm are simplicity and similarity with standard filtered back projection (FBP) along with the economic use of data.

The effect of an isolated topographic bump in a two-layer fluid on a β-plane is investigated. An analytical solution is derived in terms of the appropriate Green's function for arbitrary topography of finite horizontal extent. It is found that the disturbances generated by the bump are composed of two fundamental modes which may be wave-like or evanescent. The wave-like modes are topographically induced Rossby waves which occur only when there is eastward flow in at least one of the layers. These waves are always confined to the downstream (eastward) side of the bump. Whereas previous studies of this type have concentrated on eastward flow over topography, the theory has been extended here to include a wide range of vertically sheared flows. Particularly important is the case of low level westward flow combined with upper level eastward flow, as it has direct application, for example, to the summertime atmospheric circulation over the sub-tropical regions of the continental land.masses. In this case a wave-like disturbance extends far downstream from the bump for sufficiently large shear, and is of smaller amplitude in the upper layer than in the lower layer because of the effects of the stratification. For small shears, the wave-like mode in the lower layer is small and the character of the disturbance is evanescent, confining it to the immediate neighbourhood of the bump. A stability analysis of the solutions shows that the disturbances may be baroclinically unstable for sufficiently large mean shear.

A steady two-dimensional jet of an inviscid incompressible fluid rising and falling under gravity is considered. The jet is aimed vertically upwards and the flow is assumed to be bounded entirely by two free surfaces. The problem is solved numerically by finite differences. Accurate results for the free surface profiles are presented.

The asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem for

are investigated. Here f and g are smooth functions of position in a finite plane region with a smooth boundary. The results for the positive solution are well established, but knowledge of other branches of solutions is scarce. Here positive solutions are pieced together across lines partitioning the domain, and variational arguments are framed, as an effective means of locating the lines, so that the composite function is everywhere a solution of *. Heuristic arguments suggest strongly that there is a close relationship between the nodal lines of * and certain classes of weighted geodesic lines defined by the classical variational problem for the functional

which provides an effective basis for computation. Some results are proved but others remain conjectures. Analogous results are proved for the associated ordinary differential equation. The geometry of the solutions is surprisingly restricted when the coefficients are spatially variable. The arguments are extended to a class of reactive, diffusive systems. It is possible to predict the pattern of domains of different outcomes in terms of properties of the surface on which the reactions occur, without a knowledge of the chemical kinetics. The results appear to provide a basis for stringent testing of the postulated role of reactive-diffusive mechanisms in the formation of complex patterns in biological species.